We study the four-flavor spin model on triangular lattice using either Schwinger boson mean field or Abrikosov fermion mean field theory. Deep inside the Mott insulator, Schwinger boson theory is convenient to capture the \(120^\circ \)  magnetic order of \(\vec{S}_b\)  in the layer polarized limit with \(x=0\) . If the \(120^\circ \)  order at the bottom layer is strong, the physics is reduced to a spin \(1/2\)  boson model and we obtain either an exciton condensation phase with \(\vec{S}_t\)  ordered or a paired exciton condensation phase with \(\vec{S}_t\)  gapped. In a more interesting case, we find the \(120^\circ \)  order in the bottom layer can also be depleted by reducing the density \(n_b\) , after which we get a paired exciton condensation phase coexisting with \(Z_2\)  spin liquid. We will provide numerical evidences for spin gaps through density matrix renormalization group (DMRG) simulation. We actually find that there are two different \(Z_2\)  spin liquids with 0 or \(\pi \)  flux for the Schwinger bosons. Both 0 and \(\pi \)  flux \(Z_2\)  spin liquids have been proposed before for a single layer spin \(1/2\)  model on triangular lattice[1]}. But they are usually unstable to Schwinger boson condensation because the boson density is too large. Here because we can reduce the density of the boson in the bottom layer to \(n_b=1-x\) , it is possible to stabilize the \(Z_2\)  topological order. The \(Z_2\)  fractionalization coexists with the paired exciton condensation, and thus the phases can be called as fractional superfluids. The possibility of \(Z_2\)  spin liquids in this system opens the exciting future direction to search for superconductivity upon doping the bilayer Mott insulator, following Anderson's resonating-valence-bond (RVB) theory[2]}.
